Show the relationship between the supremum and infimum of f^2 and |f|












0














Suppose f: [a,b] $to$ $mathbb{R}$ and B satisfy |f(x)| $le$ B for every x $epsilon$ [a,b].



Show that if P = {x$_{0}$,...,x$_{n}$} is a partition of [a,b], then



M(f$^{2}$,[x$_{i-1}$,x$_{i}$]) - m(f$^{2}$,[x$_{i-1}$,x$_{i}$]) $le$
2B(M(f,[x$_{i-1}$,x$_{i}$]) - m(f,[x$_{i-1}$,x$_{i}$]))



for every 1 $le$ i $le$ n.



We are given a hint, namely that



|f(x)$^{2}$ - f(y)$^{2}$| = |f(x) - f(y)||f(x) + f(y)|.



And this has something to do with Riemann integrals, or perhaps Darboux sums, as that is the section this homework was assigned in.










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  • What are the functions $M$ and $m$?
    – Sean Roberson
    Nov 22 at 2:51










  • Those are the supremum and infimum, respectively, of the function f (or f^2) over the set in the square brackets.
    – kendal
    Nov 22 at 2:55






  • 4




    The result is obvious from the hint and triangle inequality $|f(x) +f(y) |leq 2B$ and $|f(x) - f(y) |leq M_f-m_f$
    – Paramanand Singh
    Nov 22 at 4:18
















0














Suppose f: [a,b] $to$ $mathbb{R}$ and B satisfy |f(x)| $le$ B for every x $epsilon$ [a,b].



Show that if P = {x$_{0}$,...,x$_{n}$} is a partition of [a,b], then



M(f$^{2}$,[x$_{i-1}$,x$_{i}$]) - m(f$^{2}$,[x$_{i-1}$,x$_{i}$]) $le$
2B(M(f,[x$_{i-1}$,x$_{i}$]) - m(f,[x$_{i-1}$,x$_{i}$]))



for every 1 $le$ i $le$ n.



We are given a hint, namely that



|f(x)$^{2}$ - f(y)$^{2}$| = |f(x) - f(y)||f(x) + f(y)|.



And this has something to do with Riemann integrals, or perhaps Darboux sums, as that is the section this homework was assigned in.










share|cite|improve this question
























  • What are the functions $M$ and $m$?
    – Sean Roberson
    Nov 22 at 2:51










  • Those are the supremum and infimum, respectively, of the function f (or f^2) over the set in the square brackets.
    – kendal
    Nov 22 at 2:55






  • 4




    The result is obvious from the hint and triangle inequality $|f(x) +f(y) |leq 2B$ and $|f(x) - f(y) |leq M_f-m_f$
    – Paramanand Singh
    Nov 22 at 4:18














0












0








0







Suppose f: [a,b] $to$ $mathbb{R}$ and B satisfy |f(x)| $le$ B for every x $epsilon$ [a,b].



Show that if P = {x$_{0}$,...,x$_{n}$} is a partition of [a,b], then



M(f$^{2}$,[x$_{i-1}$,x$_{i}$]) - m(f$^{2}$,[x$_{i-1}$,x$_{i}$]) $le$
2B(M(f,[x$_{i-1}$,x$_{i}$]) - m(f,[x$_{i-1}$,x$_{i}$]))



for every 1 $le$ i $le$ n.



We are given a hint, namely that



|f(x)$^{2}$ - f(y)$^{2}$| = |f(x) - f(y)||f(x) + f(y)|.



And this has something to do with Riemann integrals, or perhaps Darboux sums, as that is the section this homework was assigned in.










share|cite|improve this question















Suppose f: [a,b] $to$ $mathbb{R}$ and B satisfy |f(x)| $le$ B for every x $epsilon$ [a,b].



Show that if P = {x$_{0}$,...,x$_{n}$} is a partition of [a,b], then



M(f$^{2}$,[x$_{i-1}$,x$_{i}$]) - m(f$^{2}$,[x$_{i-1}$,x$_{i}$]) $le$
2B(M(f,[x$_{i-1}$,x$_{i}$]) - m(f,[x$_{i-1}$,x$_{i}$]))



for every 1 $le$ i $le$ n.



We are given a hint, namely that



|f(x)$^{2}$ - f(y)$^{2}$| = |f(x) - f(y)||f(x) + f(y)|.



And this has something to do with Riemann integrals, or perhaps Darboux sums, as that is the section this homework was assigned in.







real-analysis riemann-integration riemann-sum






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share|cite|improve this question













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edited Nov 22 at 3:05

























asked Nov 22 at 2:49









kendal

337




337












  • What are the functions $M$ and $m$?
    – Sean Roberson
    Nov 22 at 2:51










  • Those are the supremum and infimum, respectively, of the function f (or f^2) over the set in the square brackets.
    – kendal
    Nov 22 at 2:55






  • 4




    The result is obvious from the hint and triangle inequality $|f(x) +f(y) |leq 2B$ and $|f(x) - f(y) |leq M_f-m_f$
    – Paramanand Singh
    Nov 22 at 4:18


















  • What are the functions $M$ and $m$?
    – Sean Roberson
    Nov 22 at 2:51










  • Those are the supremum and infimum, respectively, of the function f (or f^2) over the set in the square brackets.
    – kendal
    Nov 22 at 2:55






  • 4




    The result is obvious from the hint and triangle inequality $|f(x) +f(y) |leq 2B$ and $|f(x) - f(y) |leq M_f-m_f$
    – Paramanand Singh
    Nov 22 at 4:18
















What are the functions $M$ and $m$?
– Sean Roberson
Nov 22 at 2:51




What are the functions $M$ and $m$?
– Sean Roberson
Nov 22 at 2:51












Those are the supremum and infimum, respectively, of the function f (or f^2) over the set in the square brackets.
– kendal
Nov 22 at 2:55




Those are the supremum and infimum, respectively, of the function f (or f^2) over the set in the square brackets.
– kendal
Nov 22 at 2:55




4




4




The result is obvious from the hint and triangle inequality $|f(x) +f(y) |leq 2B$ and $|f(x) - f(y) |leq M_f-m_f$
– Paramanand Singh
Nov 22 at 4:18




The result is obvious from the hint and triangle inequality $|f(x) +f(y) |leq 2B$ and $|f(x) - f(y) |leq M_f-m_f$
– Paramanand Singh
Nov 22 at 4:18















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